On normal generation of line bundles on abelian threefolds (Q1300293)

An ample line bundle \(L\) on a projective variety is called ``normally generated'' if \(L\) is very ample and the corresponding projective embedding is projectively normal. According to a theorem of Koizumi an ample line bundle of type \((d_1,\dots,d_g)\) on an abelian variety \(X\) of dimension \(g\) is normally generated if only \(d_1\geq 3\). \textit{A. Ohbuchi} [Proc. Japan Acad., Ser. A 64, No. 4, 119-120 (1988; Zbl 0699.14022)] gave a criterion for an ample line bundle of type \((2,d_2,\dots,d_g)\) on \(X\) to be normally generated. In the present paper this criterion is exploited to determine explicitly, with the exception of some geometrical situations, which line bundles of type \((2,d_2, d_3)\) on an abelian threefold are normally generated. The main tool for the proof is a result of \textit{C. Birkenhake}, \textit{H. Lange} and \textit{S. Ramanan} [Manuscr. Math. 81, No. 3-4, 299-310 (1993; Zbl 0807.14030)] on global generation of line bundles on abelian threefolds. This result is also slightly generalized.